Jane bought a new rectangular placemat with an area of 126 square inches. the length of the placemat is four times the quantity of nine less than half its width. complete the equation that models the area of the placemat, in terms of the width of the placemat, w.

Let
x-------> the length of the placemat
w-----> the width of the placemat

we know that
area of the placemat=x*w
A=126 in²
so
A=x*w-----> equation 1

x=4*(9-0.5w)-----> equation 2

substitute equation 2 in equation 1
A=4*(9-0.5w)*w------>A=36w-2w²

the equation that models the area of the placemat, in terms of the width of the placemat, w is
A=-2w²+36w

A=126 in²
126=-2w²+36w
2w²-36w+126=0

using a graph tool----> to resolve the second order equation

see the attached figure

the solutions are
w1=4.76 in
w2=13.24 in

for w1=4.76 in
x=126/w----> x=126/4.76-----> x=26.47 in

for w1=13.24 in
x=126/w----> x=126/13.24-----> x=9.52 in

the answer is
the equation that models the area of the placemat, in terms of the width of the placemat, w is
A=-2w²+36w